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Is there a p-adic introduction, focusing on the simple number theory, without prerequisite of algebra knowledge such as group/field etc?

athos
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    Even for elementary number theory we need the structure of an abelian group $(\Bbb{Z},+)$ and the field of fractions $\Bbb{Q}$. And of course, the $p$-adic numbers form a field, that is really important. So the better question is, where can I find the minimal algebraic requirements needed anyway for $p$-adic numbers? – Dietrich Burde Jul 24 '18 at 13:37
  • @DietrichBurde I agree. Actually I have a very minimal understanding of groups etc. Would like to seek some mater have no prerequisite of algebra, ie if it explains the concepts used, that would be great – athos Jul 24 '18 at 13:39
  • Then just view $p$-adic numbers via $p$-adic expansion, see wikipedia. – Dietrich Burde Jul 24 '18 at 13:41
  • @DietrichBurde wiki only explains the setup , no application — such as some useful theorems that can solve some IMO problems etc – athos Jul 24 '18 at 13:43
  • Then take any of the references given there. What applications do you have in mind? – Dietrich Burde Jul 24 '18 at 13:46
  • @DietrichBurde actually if something can help solve IMO style problems I’d be interested — I know this is probably quite trifle in academic point of view – athos Jul 24 '18 at 13:47
  • @DietrichBurde yeah :) just to learn something to have fun – athos Jul 24 '18 at 13:49
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    Then read the chapter $p$-adic numbers in the book Zahlen, by Jürgen Neukirch. I am sure there is a translation into Greek. – Dietrich Burde Jul 24 '18 at 13:50

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Some elementary references I know of are:

[1] Farhad Bill Aslan and Howard Becton Duck, The real number system vs. a system called $10$-adic, School Science and Mathematics 92 #8 (December 1992), pp. 427−432.

Summary at ERIC.

[2] Boris Mikhailovich Bekker, Sergei Vladimirovich Vostokov, and Yury J. Ionin, $2$-adic numbers, Quantum 9 #6 (July−August 1999), pp. 22−26.

A slightly revised reprint of this 1999 paper was published as Chapter 2 (pp. 99−109) in Serge Lwowitsch Tabachnikov (editor), Kvant Selecta: Algebra and Analysis, I, Mathematical World #14, American Mathematical Society, 1999. The book chapter is nearly identical except that it includes a solution (attributed to Dmitry Konstantinovich Faddeev) to the problem stated at the beginning of the section titled The $2$-adic Logarithm.

[3] Edward Bruce Burger and Thomas Struppeck, Does $\Sigma_{n=0}^{\infty}\frac{1}{n!}$ really converge? Infinite series and $p$-adic analysis, American Mathematical Monthly 103 #7 (August−September 1996), pp. 565−577.

[4] Albert [Al] Anthony Cuoco, Making a divergent series converge, Mathematics Teacher 77 #9 (December 1984), pp. 715−717.

[5] Cyrus Colton MacDuffee, The $p$-adic numbers of Hensel, American Mathematical Monthly 45 #8 (October 1938), pp. 500−508.

[6] Ilya Shevelevich Slavutskii, First steps in the geometry of $p$-adic fields, Mathematical Spectrum 28 #3 (May 1996), pp. 54−55.

Dave L. Renfro
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  • There are open access references? Reference "[1] Farhad Bill Aslan and Howard Becton Duck" is not open. Example of full open access: doi.org/10.2307/2975598. – Peter Krauss Jul 24 '23 at 11:02
  • @Peter Krauss: There are open access references? Reference "[1] Farhad Bill Aslan and Howard Becton Duck" is not open. --- My experience with posting references like this during roughly the past 25 years is that accessibility goes up over time. Everything at JSTOR was behind a paywall until the last few years -- first, they opened up things published before 1920 or so; then, sometime in the early part of COVID, they allowed "Register for free" for viewing but not downloading papers. Also, "Mathematics Teacher" ([4] above) has only been at JSTOR (continued) – Dave L. Renfro Jul 24 '23 at 17:44
  • in the last few years (off-hand, I don't know the year JSTOR acquired MT). For that matter, neither was "Mathematical Gazette" (not cited here, but often in my various literature lists posted over the years), although JSTOR took on MG a bit earlier (maybe 10-12 years ago?). Indeed, when I was posting reference lists in sci.math (whose posts I still often cite in SE comments/answers), most of what I cited was not even digitized anywhere (at least for my earlier posts -- Sept 1999 example; (continued) – Dave L. Renfro Jul 24 '23 at 17:44
  • compare this with a recent MSE post in which I was able to find a link for every paper), but knowing the existence of the papers was still useful, and I'm now pleased to find that the kinds of things I used to post can almost always be found digitized (example from 3 months ago), and even when behind a paywall, the paywall itself provides proof of the existence of the paper (shows it's not simply something I made up). (continued) – Dave L. Renfro Jul 24 '23 at 17:45
  • So while [1] may not be freely available now, it very well could be in the future. Moreover, I think lists like this, which include references from reasonably well known elementary journals (this from about 4 decades of personal "data mining" things of possible interest to me -- several thousand hours in/out of libraries looking through every issue of every volume of such journals and possibly as much time spent in photocopying; here I'm also including time with non-elementary journals for research purposes such as here) (continued) – Dave L. Renfro Jul 24 '23 at 17:46
  • can be very useful, since such papers often languish unknown to both math hobbyists and math professionals as a result of papers in such journals often not appearing in (and -- as regards to the journals for [1], [2], [4], [6] -- NEVER appearing in) MR or Zbl. – Dave L. Renfro Jul 24 '23 at 17:54